1. FIELD OF THE INVENTION
The present invention relates to the field of burst signal detection such as may be found in laser Doppler velocimetry, phase Doppler particle sizing, communications, and burst radar applications.
2. ART BACKGROUND
Reliable burst signal detection plays a very important role in the signal processing of burst signals for a wide range of applications. Specifically, in the application of the laser Doppler velocimeter (LDV) and the Phase Doppler Particle Analyzer (PDPA), short duration "Doppler burst" signals are produced when particles pass the intersection of two laser beams. These signals are characterized as consisting of a Gaussian beam intensity envelope or pedestal with a sine wave superimposed on it. The duration of the signal depends upon the particle velocity and the focused diameter of the beams at their intersection. These particle transit times can vary from milliseconds to only 100 nanoseconds or less. The amplitude of the signals will also vary over a wide range due to the range of particle sizes that may be involved in addition to the effects of passing on different trajectories through the Gaussian beam. Thus, the signal processing means must have a burst detection system that can handle a very wide dynamic range and that is very fast in its response.
Added to the difficulty of detecting the burst signal is the usual problem of low signal to noise ratio (SNR). In general, the LDV and PDPA will need to detect low amplitude signals and signals that have a very low SNR (as low as 0 dB). The SNR will also vary during the measurement routine and in the past, this has entailed careful setting of the detection level or threshold value, when operating the instruments.
Burst detectors that are based upon the signal amplitude and referred to generically as "time domain" burst detectors are frequently used. See, Van Tree H., "Detection, Estimation and Modulation Theory", New York, (Wiley 1968); Ibrahim K., Werthimer D. and Bachalo W., "Signal Processing Considerations for Laser Doppler and Phase Doppler Applications", Proceedings of the Fifth International Sumposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, July 1990; Bachalo W., Werthimer D., Raffanti R. and Hermes R. "A High Speed Doppler Signal Processor for Frequency and Phase Measurement", Third International Conference on Laser Anemometry, Advances and Applications, September 1989. These devices act on the unfiltered signal or the high pass filtered signal and base the detection on the condition that the signal voltage must exceed a preset level (threshold) before the signal is considered present. A more advanced version of this method requires that each cycle in the burst signal cross a positive threshold voltage, pass through zero to a negative threshold voltage, and then the next zero voltage crossing of the signal is used in the signal period determination. However, this sequence must be repeated for each cycle in the burst for it to be accurately detected.
Time domain burst detectors have also used the average of the signal power over a certain period of time. In this approach, the signal is rectified or squared to maximize the signal amplitude while minimizing the relative affect of the noise. Unfortunately, these methods fail to work when the noise power becomes comparable to the signal power (i.e., SNR is close to 0 dB).
An optimum configuration for signal detection can be attained by using a correlation receiver in which the input signal r(t) consists of the signal component s(t) carrying the desired information plus noise n(t). In the absence of the desired signal, the input signal will only be the noise. Thus, with this configuration, the input signal r(t) is represented as either: EQU r(t)=s(t)+n(t) (1)
when the signal s(t) exists, or EQU r(t)=n(t) (2)
for the case when the signal is absent.
The signal r(t) is correlated with a stored replica of the signal s(t) to be detected in the presence of noise n(t). In this special case, for the sake of example, the signal characteristics are known. The correlator output is then compared with a certain threshold to provide a decision for the existence or non existence of the signal s(t).
For laser anenometry, the signal s(t) could be a member of an orthogonal set of sinusoidal waveforms (that is, the correlation between two sinusoidal waves are zero unless the frequencies are equal). To cover the range of frequencies of interest requires that the signal be correlated with a predetermined number of signals, m, in the manner described for the single signal. A burst detector configuration will include a correlator for correlating the signal with a set of sinusoidal waveforms of different frequencies. The maximum of the correlator outputs is then selected and compared with a certain threshold to decide whether or not the signal exists.
For example, in one implementation, the input signal is sampled with an analog to digital convertor (ADC). This changes the continuous signal to a set of discrete samples. A discrete Fourier Transform (DFT) is then performed. The series of sinusoidal waves s.sub.m (t) are correlated with the input signal by taking discrete sampled values of s.sub.m (t) represented as s.sub.m (i). The index i will range from 0 to N-1, where N is the number of samples acquired over the Discrete Fourier Transform.
For the evaluation of the burst detector, two metrics of merit are introduced. The first one is the acceptance rate (A). This metric is introduced for system evaluation in the presence of the signal (i.e., r(t)=s(t)+(t)). It is defined as the probability of signal detection provided that the signal exists. The second metric is the false detection rate (R). This metric is introduced for system evaluation in the absence of the signal. It is defined as the probability of false detection (assuming that the signal does not exist (i.e., r(t)=n(t)).
In the following, the performance of the above configuration that uses the discrete sampled data is analyzed. For the purpose of the analysis, let r(i) (where 0.ltoreq.i&lt;N) represent the discrete sampled data for the signal r(t). Thus, the discrete samples r(i) can be written as EQU r(i)=s(i)+n(i) (3) EQU r(i)=n(i) (4)
for the case of signal existence or absence respectively.
The DFT for discrete sampled data is then given by ##EQU1## where 0.ltoreq.k,i&lt;N.
The power at the k-th bin is then given by: ##EQU2##
Two expressions for S(k) will be derived. The first is derived when the signal does not exist (i.e., r(i)=n(i)). The second is derived when the signal exists (i.e., r(i)=s(i)+n(i)). These two expressions will then be used to compute the acceptance and the false detection rates.
Consider first, the case of the signal absence (i.e., r(i)=n(i)). For this case equation (6) becomes, ##EQU3##
Assuming that the n(i) and n(j) to be uncorrelated for i.noteq.j (i.e., the noise is white) then using the central limit theorem, the terms ##EQU4## can be considered to be a Gaussian random variable. Let the mean and the variance of n(i) respectively be 0 and .sigma..sup.2 (i.e. the noise is of zero mean and power of .sigma..sup.2). The mean and the variance of the terms Q(k) and C(k) will therefore be zero and N.sigma..sup.2 /2, respectively. Thus, the term S(k) or equation (7) can be considered to be a random variable of chi square statistic with 2 degrees of freedom. Furthermore, the probability density function for S(k) is given by: ##EQU5## where y.gtoreq.0.
For a certain threshold Th in the power spectra, the probability that any of the S(k)'s below that threshold is then given by: ##EQU6## Thus, the probability that all the S(k)'s do not exceed that threshold is then given by: ##EQU7## The false detection rate can be then written as: ##EQU8## For (Th/No.sup.2) greater than 1, the above equation can be approximated to: ##EQU9##
As it can be seen, the rejection rate R(Th) is dependent on the noise level. By changing the noise power, the threshold should be changed to maintain the same rejection rate.
In the following discussion, the relationship for finding the power spectra for the case of signal existence is established. For this case, r(i)=s(i)=n(i), and equation(6) can be written as: ##EQU10##
Let s(i) be equal to M sin(2.pi.im/N), (i.e., the signal is a discrete sampled sinusoidal wave of frequency mf.sub.s /N where f.sub.s is the sampling frequency). Thus, the signal frequency corresponds to the discrete frequency m in the DFT), and the power at the m-th bin is given by: ##EQU11##
The above equation can be simplified as: ##EQU12##
For the case where ##EQU13## (this condition is generally satisfied for low SNR less than 10 dB), the above equation can be rewritten as: ##EQU14##
Therefore, S(m) can be approximated by a d.c. value (MN).sup.2 /4 added to the second term. As it is shown earlier, the second term can be considered as a random variable of chi square distribution given by: ##EQU15##
Thus, it is shown that the power spectra of the noise will have a chi squared distribution. A formula for the false detection of noise was derived (equation(13)). Then it was shown that for the case of the signal plus noise (r(t)=s(t)+n(t)), the power spectra was composed of two components (equation(17)). The first component is due to the power spectra of the signal without the noise whereas the second was due to the noise and had a chi squared distribution. In general, the power in the bin corresponding to the signal frequency is greater than (MN).sup.2 /4. An initial crude selection of a signal detection threshold Th in the power spectra to be a value equal to (MN).sup.2 /4 would ensure signal detection. With this value substituted into equation (13), the false detection rate would be given by: ##EQU16##
Although the above method provides optimum performance for signal detection, it is an impractical method for two reasons.
First, it requires intensive computation. Thus its implementation requires complex circuitry and is only limited to low frequency with long burst applications. In addition, the threshold selection for optimum performance is dependent on the signal level and the noise level. Thus, the threshold should be changed every time any of these parameters is changed.
In the present invention, a frequency domain burst detector will be described. The burst detector is capable of detecting signals at very low SNR, utilizing minimal computational overhead, and does not require frequent settings of a threshold level which is set based on the signal to noise ratio in the frequency domain rather than on the amplitude.